Cluster sample inference using sensitivity analysis: the case with few groups Johan Vikström WORKING PAPER 2009:15 The Institute for Labour Market Policy Evaluation (IFAU) is a research institute under the Swedish Ministry of Employment, situated in Uppsala. IFAU’s objective is to promote, support and carry out scientific evaluations. The assignment includes: the effects of labour market policies, studies of the functioning of the labour market, the labour market effects of educational policies and the labour market effects of social insurance policies. IFAU shall also disseminate its results so that they become acces sible to different interested parties in Sweden and abroad. IFAU also provides funding for research projects within its areas of interest. The deadline for applications is October 1 each year. Since the researchers at IFAU are mainly economists, researchers from other disciplines are encouraged to apply for funding. IFAU is run by a Director-General. 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The purpose of the Working Paper Series is to provide a factual basis for public policy and the public policy discussion. ISSN 1651-1166 Cluster sample inference using sensitivity analysis: the case with few groupsa by Johan Vikstr¨omb 11th June, 2009 Abstract This paper re-examines inference for cluster samples. Sensitivity analysis is proposed as a new method to perform inference when the number of groups is small. Based on estimations using disaggregated data, the sensitivity of the standard errors with respect to the variance of the cluster effects can be examined in order to distinguish a causal effect from random shocks. The method even handles just-identiﬁed models. One important example of a just-identiﬁed model is the two groups and two time periods difference-in differences setting. The method allows for different types of correlation over time and between groups in the cluster effects. Keywords: Cluster-correlation; Difference-in-Differences; Sensitivity analysis. JEL-codes: C12; C21; C23. aI would like to thank Per Johansson, Michael Svarer, Nikolay Angelov, Xaiver de Luna, Gerard van den Berg, Bas van der Klaauw, Per Petterson-Lidbom, and seminar participants at VU-Amsterdam, IFAU ¨ Uppsala, Orebro University, Stockholm University, and the RTN Microdata meeting in London for helpful comments. All remaining errors are mine. This paper was initiated when I visited University of Southern California supported by the Tom Hedelius Foundation and written in part when I was employed by VU Amsterdam. The ﬁnancial support of the Swedish Council of Working Life and Social Research FAS (dnr 2004-2005) is acknowledged b IFAU and Uppsala University, [email protected] IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 1 Table of contens 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Basic model and bias in the regular OLS standard errors . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Sensitivity analysis for cluster samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 4.1 4.2 Extended sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Correlation over time in the cluster effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Multi-way clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 5.1 5.2 Monte Carlo evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Small sample properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Robustness and comparison with other inference methods . . . . . . . . . . . . . . . . . . . . . . . . 19 6 6.1 6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Application 1: Disability beneﬁts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Application 2: Earned income tax credit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 1 Introduction In many studies the analysis sample consists of observations from a number of groups, for example families, regions, municipalities, or schools. These cluster samples impose inference problems, as the outcomes for the individuals within the groups usually cannot be assumed to be independent. Moulton (1990) shows that such intra-group correlation may severely bias the standard errors. This clustering problem occurs in many difference in-differences (DID) settings, where one usually use variation between groups and over time to estimate the effect of a policy on outcomes at the individual level. As such the DID methodology is compelling, since it has the possibility of offering transparent evidence, which is also reﬂected in the exploding number of studies using the approach, for surveys see e.g. Meyer (1995) and Angrist & Krueger (2000). Many of these studies use data from only a small number of groups, such as data for men and women, a couple of states, or data from only a few schools or villages. For more examples see e.g. Ashenfelter & Card (1985), Meyer et al. (1995), Card & Krueger (1994), Gruber & Poterba (1994), Eissa & Liebman (1996), Imbens et al. (2001), Eberts et al. (2002), and Finkelstein (2002). The purpose of this paper is to provide a new method of performing inference when the number of groups is small, as is the case in these studies. The importance of performing correct inference is also reﬂected in the growing num ber of studies addressing the inference problem.1 One key insight from this literature is that the number of groups is important when deciding how to address the clustering problem. If the analysis sample consists of data from a larger number of groups, several solutions to the inference problem are available; the cluster formula developed by Liang & Zeger (1986), different bootstrap procedures (see e.g. Cameron et al. (2008)), or para metric methods (see e.g. Moulton (1990)). As expected however several Monte Carlo studies show that these methods perform rather poorly if the number of groups is small.2 1 See e.g. Moulton 1986, 1990, Arrelano (1987), Bell & McCaffrey (2002), Wooldridge 2003, 2006 Bertrand et al. (2004), Kezdi (2004), Conley & Taber (2005), Donald & Lang (2007), Hansen2007a, 2007b, Ibragi mov & Muller (2007), Abadie et al. (2007) and Cameron et al. (2008). Related studies are Abadie (2005) and Athey & Imbens (2006) which study semi-parametric and non-parametric DID estimation. 2 See e.g. Bertrand et al. (2004), Donald & Lang (2007), Cameron et al. (2008), Ibragimov & Muller (2007), and Hansen (2007a). IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 3 To address this problem Donald & Lang (2007) introduce a between estimator based on data aggregated at group level.3 They show that under certain assumptions, the aggre gated error term is i.i.d normal and standard normal inference can be applied even if the sample consists of data from a small number of groups. Their method works as long as the number of groups is not too small. Since their method is based on aggregated data their inference will be conservative in the absence of within group correlation, or if the within group correlation is small. In the limit case when the model is just-identiﬁed, i.e. when the number of aggregated observations equals the number of variables varying at group level it not possible to perform Donald & Lang (2007).4 An important example of a just-identiﬁed model is the two groups and two time periods DID setting. Another alter native is the two-stage minimum distance approach suggested by Wooldridge (2006). One important by-product of this approach is a simple test for the presence of within cluster correlation. However, as for the Donald & Lang (2007) approach the test does not work if the model is just-identiﬁed, as it is then based on a chi-square statistic with zero degrees of freedom. A ﬁnal alternative is to use bias corrected standard errors as suggested by Bell & McCaffrey (2002). The method has two limitations; it does not work if the number of groups becomes too small or if the model includes a dummy variable taking the value one for exactly one cluster and zero otherwise. As a response this paper proposes to use sensitivity analysis as a new method of per forming inference when the number of groups is small. Design sensitivity analysis has traditionally been used to test whether an estimate is sensitive to different kinds of selec tivity bias: see e.g. Cornﬁeld et al. (1959) and Bross (1966), further see e.g. Rosenbaum & Rubin (1983), Lin et al. (1998), Copas & Eguchi (2001), Imbens (2003), Rosenbaum (2004) and de Luna & Lundin (n.d.). In these papers sensitivity analysis is performed with respect to the unconfoundedness assumption or with respect to the assumption of random missing data. If these assumptions hold, the usual estimators are unbiased and the sensitivity analysis amounts to assessing how far one can deviate from for example the unconfoundedness assumption before changing the estimate by some pre-speciﬁed 3 Under 4 The 4 certain assumptions the aggregation can be made on group-time level, instead of group-level. inference is then based on a t-statistic with zero degrees of freedom. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups amount. My sensitivity analysis approach is similar, but nevertheless different in spirit. Under the assumption of no within group correlation standard normal i.i.d. inference based on disaggregated data is applicable. If this assumption is violated any standard errors based on the assumption of no within group correlation will be biased downwards. It is shown that under certain assumptions this bias can be expressed in terms of a few parameters, called sensitivity parameters. In the basic case the variance is expressed in terms of a single sensitivity parameter, deﬁned as the ratio between the variance of the group com mon error term creating within cluster correlation, and the variance of the individual error term. The sensitivity analysis then amounts to assessing how much one can deviate from the assumption of no within group correlation before changing the standard error estimate by some pre-speciﬁed amount. That is to investigate how sensitive the standard errors are to within group correlation. The test can also be inverted in order to calculate a cut-off value, where higher values of the sensitivity parameter or simply larger variance of the group common shocks renders a certain estimate insigniﬁcant. If this cut-off value is un reasonably large one can be conﬁdent that the null hypothesis of no effect can be rejected. Optimally one could use information from other sources, for instance data from other countries, other time periods, or for another outcome, in order to assess the reasonable size of the sensitivity parameter. The approach proposed in this paper is therefore similar to standard sensitivity analysis, since it also assesses how much one can deviate from an important assumption, but it is also different in spirit since it is performed with respect to bias in the standard errors and not with respect to bias in the point estimate. One key question is of course how to assess whether the sensitivity cut-off value is unreasonably large, that is how to assess the reasonable size of the within group correla tion. I believe that this has to be done on a case by case basis. However, one advantage with the approach here is that the basic sensitivity parameter is deﬁned as a ratio between two variances. It gives a sensitivity parameter with a clear economic interpretation, which of course is a basic condition for an informative sensitivity analysis. The next step is the discussion about a reasonable size of the sensitivity parameter. In order to shed more light on this issue two applications are provided. The sensitivity analysis method is applied to IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 5 data analyzed in Meyer et al. (1995) on the effects of an increase in disability beneﬁts on the duration of the period spent out of work and to Eissa & Liebman (1996) on the effects of an expansion in the earned income tax credit on labor supply. In both these studies key regressions are based on just-identiﬁed models. The sensitivity analyses indicate that the conclusion from the ﬁrst study that the treatment effect is signiﬁcant is not sensitive to departure from the independence (no-cluster) assumption, whereas the results of the second study are sensitive to the same departure and its conclusion cannot therefore be trusted. It demonstrates that the sensitivity analysis approach is indeed helpful for determining the validity of treatment effects. By introducing sensitivity analysis in this way, this paper contributes in several ways. The method is applicable when the analysis sample consists of data from only a small number of groups. It even handles just-identiﬁed models. As no other method is applica ble in the just-identiﬁed case it is the best application of the sensitivity analysis method. If the model is not just-identiﬁed but the number of groups is still small, the Monte Carlo study in this paper show that the sensitivity analysis method offers an attractive alternative compared to other commonly used methods. The method is also able to handle different types of correlation in the cluster effects, most importantly correlation within the group over time and multi-way clustering. This is done by introducing several sensitivity pa rameters. The paper is structured as follows. Section 2 presents the basic model and analyzes the asymptotic bias (asymptotic in the number of disaggregated observations) of the OLS standard errors. Section 3 introduces the basic sensitivity analysis approach. Section 4 extends these basic results to more general settings. It is shown that different assumptions about the cluster effects lead to different types of sensitivity analyses. Section 5 presents Monte Carlo estimates on the performance of the sensitivity analysis method. The method is also compared to other commonly used methods of performing inference. Section 6 presents the two applications, and Section 7 concludes. 6 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 2 Basic model and bias in the regular OLS standard errors Consider a standard time-series/cross section model. Take a linear model for the outcome y for individual i in time period t in group g as � β + eigt yigt = xigt (1) eigt = cgt + εigt Here εigt is an individual time speciﬁc error, cgt is a cluster effect which varies across groups and time, and xigt the regressors. Of course individuals can represent any disag gregated unit. The regressors may or may not include ﬁxed group effects and/or ﬁxed time effects. This model covers a wide range of different models, including a ”simple” cross-section, with data from for instance a couple of schools or villages. Another impor tant example is the heavily used standard DID model. In a regression framework, a usual DID model is yigt = αg + αt + bDgt + cgt + εigt , (2) including ﬁxed time, αt , and ﬁxed group effects, αg , and where Dgt is an indicator func tion taking the value one if the intervention of interest is implemented in group g at time point t and zero otherwise. The treatment effect is hence identiﬁed through the variation between groups and over time. In this setting cgt can be given a speciﬁc interpretation as any group-time speciﬁc shocks.5 Deﬁne N = ∑G ∑T ngt , where G is the number of groups, T is the number of time periods, and ngt is the number of individual observations for group g in time period t. If E[eigt |xigt ] = 0, the ordinary least square (OLS) estimate of β β� = (X � X)−1 X �Y (3) 5c gt also captures any differences in the group mean due to changes in the composition of the group over time. If ngt is large this problem is mitigated. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 7 is an unbiased estimate of β . Here Y is a N-vector collecting all yigt , X is a N × K matrix containing the observations of the independent variables, and accordingly β a K-vector of the coefﬁcients of interest. Next consider inference. Assume that E(ee� ) = σ 2C, where e is a N-vector collecting all eigt , and σ 2 ≡ 1/N tr(ee� ), and C is a positive-deﬁnite matrix that captures the correlation in the error terms between the individuals. The true covariance matrix is then V = σ 2 (X � X)−1 X �CX(X � X)−1 , (4) which can be compared with the regular OLS covariance matrix formula Vˆ = σˆ 2 (X � X)−1 . (5) The asymptotic bias (asymptotic in the number of individuals (N)) of the regular stan dard errors has been analyzed extensively: see e.g. Greenwald (1983): other contributions are Campbell (1977), Kloek (1981) and Holt & Scott (1982). To be clear here we mean asymptotic in the number of individuals (N). Following equation (9)-(11) in Greenwald (1983) and some algebraic manipulations6 gives the asymptotic bias in the estimated co variance matrix which can be expressed as E(Vˆ ) −V = σ2 (6) � � tr[(X � X)−1 X � (I −C)X] � −1 (X X) + (X � X)−1 X � (I −C)X(X � X)−1 . N −K Hence if C = I, that is the identity matrix, the estimated covariance matrix is an unbiased estimate of the true covariance matrix, and the estimated standard errors are unbiased. It 6 Notice that V and n are deﬁned in a different way here compared to Greenwald (1983). The expression fol lows from substituting equation (10) and (11) in Greenwald (1983) into equation (9) in Greenwald (1983), breaking out σ 2 and simplifying. 8 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups holds if cgt = 0 for all g and all t, and if εigt is i.i.d. This general formula incorporates the two main reasons for bias in the standard errors into one expression. They are: (i) the cluster correlation problem caused by the presence of cgt , highlighted by Moulton (1990), and (ii) the policy autocorrelation problem caused by correlation over time in cgt , highlighted by Bertrand et al. (2004). The exact size of these problems depend on the case speciﬁc shape of C. For the model in equation (1) the bias is negative, i.e. V is larger than E(Vˆ ). It should also be noted that the bias consist of two distinct parts. Thus in the case of cluster effects in form of within group correlation the OLS standard errors underestimate the true standard errors. First, the OLS estimator of the error variance σˆ 2 , is neither an unbiased nor a consistent estimator of the true error variance σ 2 , if the error covariance matrix does not satisfy the OLS assumptions. Second, and more obvious, even if the error variance is known, the standard errors are biased since the coefﬁcient covariance matrix is mis speciﬁed. 3 Sensitivity analysis for cluster samples The discussion in the previous section reveals that whether or not cgt = 0 is crucial for how to perform inference. If cgt = 0 regular OLS inference can be performed, possibly with control for heteroscedasticity. If cgt �= 0 on the other hand the regular OLS stan dard errors will be severely biased. As shown by Donald & Lang (2007) this has very important implications when the number of groups is small. They introduce a between estimator based on data aggregated at group level. It creates an all or nothing situation; under the assumption of cgt = 0 there are apparently narrow conﬁdence intervals based on individual data, and under the assumption of cgt �= 0 there are apparently very wide conﬁ dence intervals based on aggregated data. Needless to say arguing that cgt = 0 will almost always be very difﬁcult, whereas arguing that the variance of cgt is small is reasonable in many applications. In the end it is the size variance of the within group correlation that matters. This is the key idea behind the new method proposed in this paper. Formally, the starting point for the sensitivity analysis method is the general formula IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 9 for the bias in the regular OLS standard errors presented in equation (6). This expression is based on derivations in Greenwald (1983), which among other expressions uses E(σˆ 2 ) = σ 2 (N − tr[(X � X)−1 X �CX])/(N − K).7 (7) Combining this expression with the deﬁnition of V in equation (4) and the deﬁnition of Vˆ in equation (5), and noting that E(Vˆ ) = E(σˆ 2 )(X � X) gives V= N −K (X � X)−1 X �CXE(Vˆ ). N − tr[(X � X)−1 X �CX] (8) Further plim(Vˆ ) = E(Vˆ ) and thus E(Vˆ ) can be consistently (in terms of number of indi viduals) estimated by Vˆ . Starting with this equation the idea behind the sensitivity analysis is straightforward. Faced with a cluster sample with data from only a small number of groups, we can use disaggregated data and estimate β using OLS. Then estimate Vˆ in equation (4) as if there were no cluster effects. Then notice that Vˆ only gives correct standard errors if cgt = 0 for all g and t. However, as a sensitivity analysis we can use the expression above and express the bias in the covariance matrix in terms of different so called sensitivity parameters, and assess how large they have to be in order to change the variance of a parameter estimate by a certain amount: that is, if the results are insensitive to departures from the assumption of no within group correlation, it indicates that the results can be trusted. As shown below the exact speciﬁcation of the sensitivity parameters will depend on the assumptions which can be imposed on C. Let us start with the simplest case. If ε is homoscedastic and if E(cgt cg�t ) = 0 for � t � , then the full error term, all t and all g �= g� , and E(cgt cgt � ) = 0 for all g and all t = eigt = cgt + εigt , is homoscedastic8 , equi-correlated within the group-time cell and uncor related between the group-time cells. Further assume ngt = n and xigt = xgt , that is, the 7 See derivations of equation (A.3) in Greenwald (1983). sensitivity analysis throughout this paper is made under the homoscedasticity assumption. The as sumption makes it possible to write the bias in terms of single parameters. If one suspect heteroscedasticity, one approach is to use standard errors robust to heteroscedasticity in the spirit of White (1980), and use this covariance matrix instead of Vˆ . The sensitivity analysis based on this speciﬁcation will then be conservative. 8 The 10 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups regressors are constant within each group, and constant group size. This special case has been analyzed by Kloek (1981).9 He shows that under these assumptions equation (8) reduces to nGT − K V = E(Vˆ )τ nGT − Kτ (9) with τ = 1 + (n − 1) σc2 . σc2 + σε2 (10) Here σc2 is the variance of c, and σε2 the variance of ε. Expressing the ratio between these two variances as σc2 = γσε2 gives � � γ nGT − K V = E(Vˆ ) 1 + (n − 1) 1 + γ nGT − K(1 + (n − 1) 1+γ γ ) (11) In other words the bias in the covariance matrix is expressed in terms of observables and a single unknown parameter γ, which is interpreted as the relation between the variance of the group-time error term and the variance of the individual error term.10 Using standard textbook results; if γ = 0, that is if there is no within group correlation, and ∑ n jt is large βˆa t=� a ∼ N(0, 1), (12) E(Vˆaa ) where βˆa is the ath element of βˆ , and Vˆaa the element in the ath column and ath row of Vˆ . Furthermore if γ �= 0 and known, c jt ∼ N(0, σc2 )11 , and ∑ n jt is large βˆa βˆa t=√ =� γ Vaa nGT −K E(Vˆaa )(1 + (n − 1) 1+γ ) nGT −K(1+(n−1) a ∼ N(0, 1). (13) γ 1+γ ) 9 Kloek (1981) analyzes the one dimensional case with only a group dimension and no time dimension. A group-time version of his proof is presented in Appendix. 10 Actually γ is only potentially unknown. If the number of groups is larger σ 2 can be consistently estimated c using the between group variation, and σε2 can be consistently estimated using the within group variation, and this gives p. 11 The normality assumption can be replaced by any other distributional assumption, for instance a uniform distribution. However this will complicate the sensitivity analysis, since the combined error term will have a mixed distribution. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 11 It is then possible to use γ as a sensitivity parameter. After estimating βˆa and con sistently estimating E(Vˆaa ) by Vˆaa using the disaggregated data, the sensitivity analysis then amounts to assessing how much γ has to deviate from zero in order to change the standard errors by a pre-speciﬁed amount. The sensitivity analysis method is applicable as long as the model is identiﬁed. In the present case with variables constant within each group-time cell, this holds if GT ≥ K, i.e. if the number of group-time cells is larger than or equal to the number of explanatory variables. In other words our sensitivity analysis method even handles just-identiﬁed models, for instance the two groups and two time periods DID setting. As no other method is applicable in the just-identiﬁed case it is the best application of the sensitivity analysis method. If the model is not just-identiﬁed but the number of groups is still small the sensitivity analysis method offers an alternative to other commonly used methods such as the Donald & Lang (2007) approach. The test can also be inverted in order to calculate the γ value which corresponds to a speciﬁc p-value. One could for example be interested in the γ cut-off value which renders the estimated treatment effect statistically insigniﬁcant at α% level. This follows from setting t = Z1−α/2 and solve for γ in the equation (13) above γc,a = 2 Vˆaa )(nGT − K) (βˆa2 − Z1−α/2 . Vˆaa )(nGT − K) − βˆ 2 (nGT − nK) 2 (nZ1−α/2 (14) a Here Zυ is the υ quantile of the standard normal distribution. Note that we have replaced E(Vˆaa ) with Vˆaa as it is consistently estimated by Vˆaa . Furthermore, note that γc,a depends on n, the number of observations for each group. This dependence comes both from Vˆ which decreases as n increases and also directly as n enters the expression for γc,a . Taken together these two effects means that γc,a increases as n goes from being rather small to moderately large: however as n becomes large this effect ﬂattens out, and γc,a is basically constant for large n. If γc,a is unreasonably large, one could be conﬁdent that the null-hypothesis about zero effect could be rejected. The key question then becomes: what is unreasonably large? At the end of the day, as with all sensitivity analyses, some judgment has to be made. Since 12 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups the true γ may vary a lot between different applications, we believe that the assessment has to be done on a case by case basis. However, the sensitivity analysis presented here avoids the common sensitivity analysis pitfall. That is, that one is left with a sensitivity parameter which is hard to interpret and thus hard to relate to economic conditions. Here the basic sensitivity parameter, γ, is deﬁned as the ratio between two variances, which makes it both easier to interpret and easier to discuss. Optimally one could also use information from other sources to make the discussion more informative for instance data from another country, other time periods, or for another outcome. In some cases it may also be beneﬁcial to re-scale γ. The two applications presented in Section 6 using data from Meyer et al. (1995) and Eissa & Liebman (1996) further exemplify how γ can be interpreted. If either the assumption of either ngt = n or xigt = xgt is relaxed the sensitivity analysis is still straightforward. Note that the general formula for the bias presented in equa tion (8) nevertheless holds. In the basic case with ngt = n or xigt = xgt this expression could be simpliﬁed considerably. In general under assumption E(cgt cg�t ) = 0, assump tion E(cgt cgt � ) = 0, and with the model speciﬁed as in equation (1), C has the familiar block-diagonal structure ⎡ ⎤ C ... 0 ⎥ ⎢ 1 .. ⎥ ⎢ .. . . C = ⎢ . . . ⎥ ⎦ ⎣ 0 . . . CGT (15) with CGT = [(1 − 1+γ γ )Igt + 1+γ γ Jgt ]. Here IGT is an ngt times ngt identity matrix, and Jgt is an ngt times ngt matrix of ones: γc,a is then found by numerically solving for γ in βˆa Z1−α/2 = √ , Vaa (16) with V deﬁned as in equation (8) and C deﬁned as in equation (15) above. From calcu lations I note that in general γc,a is quite insensitive to violations of ngt = n, except when some groups are very large and others are very small. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 13 4 Extended sensitivity analysis 4.1 Correlation over time in the cluster eﬀects The sensitivity analysis presented in the previous section is applicable under a num � g� , and ber of assumptions about cgt . Most notably E(cgt cg�t ) = 0 for all t and all g = � t � . In many studies E(cgt cgt � ) = 0 for all g is a re E(cgt cgt � ) = 0 for all g and all t = strictive assumption. In a model with ﬁxed group and ﬁxed time effects, cgt captures any group-time shocks. Consider a study on the effects of minimum wages on employment using variation across regions and over time. The group-time shocks then capture all re gional speciﬁc shocks in employment. If present they are most likely correlated over time. This problem, often refereed to as the policy autocorrelation problem, was highlighted by Bertrand et al. (2004). This subsection therefore relax the assumption that E(cgt cgt � ) = 0: instead we assume an AR(1) structure for cgt cgt = κcgt−1 + dgt , (17) where dgt is assumed to be a white noise series with mean zero and variance σd2 . Further, assume that |κ| < 1. I make the natural extension of the basic sensitivity analysis and deﬁne σd2 = γσε2 . It gives two sensitivity parameters, γ and κ, instead of the single sensi tivity parameter γ. Then if κ = 0 the basic sensitivity analysis is applicable. To be clear, κ is interpreted as the ﬁrst-order autocorrelation coefﬁcient for cgt , and γ as the relation between the variance of the group-time speciﬁc shock and the variance of the unobserved heterogeneity. Consider the case with repeated cross-section data. Assume that data on ngt individu als from group g in time period t are available. The general formula presented in equation (8) for the covariance matrix still hold. However, since cgt is allowed to follow an arbi trary AR(1) process, C will obviously differ from the basic sensitivity analysis. In order to express C in terms of κ and γ we use the well know properties of an AR(1) process. It turns that out if ngt = n and xigt = xgt holds, there is a simple expression for the relation 14 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups between V and Vˆ Vaa ≈ E(Vˆaa )(1 + (n − 1) γ γ +n Haa ) 2 1 + γ − κ2 1+γ −κ (18) where Haa is the element in the ath column and ath row of H given by � � −1 � H = ( ∑ ∑ xgt xgt ) ∑ ∑ ∑ (κ |t−t | xgt xgt �) g t g t t � =t � The proof can be found in Appendix. Based on this simple expression for the bias in the regular OLS standard errors, one can assess the sensitivity of the standard errors with respect to both the autocorrelation and the variance of the group-time speciﬁc shocks. As for the basic sensitivity analysis one may be interested in the cut-off value which renders an interesting estimate insigniﬁcant. In this case with two sensitivity parameters a natural way to proceed is to solve for γ for a range of values of κ. Let us that interest lies in the effect of variable a, then the cut-off value for γ is 2 V ˆaa )(1 − κ 2 ) (βˆa2 − Zα/2 γc,a = . (nZ 2 Vˆaa )(1 + Haa ) − βˆ 2 α/2 (19) a Again note that E(Vˆaa ) is replaced with Vˆaa as it is consistently estimated by Vˆaa . If the combinations of γc,a and κ values are unreasonable large, one could be conﬁdent in that the null hypothesis about zero effect should be rejected. Also note that γc,a can either increase or decrease with κ, as Haa can either increase or decrease with κ. If either ngt = n or xigt = xgt do not hold it is not possible to obtain a closed from solution for γc,a . But using numerical methods, it is possible to solve for γ in βˆa Z1−α/2 = √ , Vaa (20) for a range of values of κ and the desired signiﬁcance level. Here V is deﬁned in equation (8), and C is deﬁned in equation (A.13) presented in appendix. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 15 4.2 Multi-way clustering Consider an application where we have data from a number of regions and where the re gion is deﬁned as the group. In the sensitivity analysis presented so far, the assumption of E(cgt cg�t ) = 0 is crucial. In other words it is assumed that the outcomes for individ uals within a region are correlated and that there is no correlation between individuals on different sides of the border between two different regions. Most likely this will be violated in many applications. Here this assumption is relaxed in the situation with cross section data. Assume that the groups can be divided into group clusters containing one or more groups. Dropping the time dimension, the outcome y for individual i in group g in group-cluster s is yigs = xigs β + cgs + εigs . (21) Retain the deﬁnition of γ from the basic sensitivity analysis as σc2 = γσε2 . γ is then again interpreted as the relation between the variance of the group-time shocks and the variance of the individual unobserved heterogeneity. Further assume that if s �= s� then E(cgs cg� s� ) = 0 and if s = s� then E(cgs cg� s� ) = ξ σc2 .12 ξ should be interpreted as the relation between the inter-group correlation and the intra-group correlation for groups in the same cluster of groups. This means that it will be far below one in many applications. Note that the general expression for the covariance matrix presented in equation (8) holds. If the above assumptions hold, and if ng = n and xigt = xgt hold, the derivations in the appendix show that there is a simple relation between Vaa and Vˆaa Vaa ≈ Vˆaa (1 + (n − 1) γ γ +n ξ Maa ) 1+γ 1+γ (22) where Maa is the element in the ath column and ath row of M given by � −1 M = ( ∑ ∑ xgs xgs ) ∑∑ s g s (xgs xg� � s ). ∑ � g g �=g 12 It is obviously possible to also allow for an time-dimension, which generally gives sensitivity analysis in three parameters, which would measure the variance, the autocorrelation respectively the between group correlation in the cluster effects. 16 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups Again there are two sensitivity parameters, γ and ξ . As in the previous case one can proceed to solve for γc,a for a range of values of ξ . Let us that the interest lies in the effect of variable a: then γc,a = 2 V ˆaa βˆa2 − Zα/2 (nZα2 /2Vˆaa )(1 + ξ Maa ) − βˆa2 . (23) If these combinations of γc,a and ξ values are unreasonable large, one could be conﬁdent that the null hypothesis about zero effect should be rejected. One could also interpret the division of the groups into group clusters as a sensitivity analysis. The standard errors may be sensitive to some divisions but not to others. Note that introducing multi-way clustering in the way done here increases the standard errors, and thus γc,a decreases with ξ. If either ngt = n or xigt = xgt do not hold it is not possible to obtain a closed from solution for γc,a . But it is possible to solve for γ in βˆa Z1−α/2 = √ , Vaa (24) for a range of values of ξ and the desired signiﬁcance level. Here V is deﬁned in equation (8), and C is deﬁned in equation (A.25) presented in the appendix. 5 Monte Carlo evidence This section provides Monte Carlo estimates of the performance of the proposed sensi tivity analysis method. The small sample properties of the method and the sensitive of method is to the choice of reasonable γ are investigated. The sensitivity analysis method is also compared to other commonly used inference methods. I consider a DID set up. The treatment is assumed to vary at group-time level, and the interest lies in estimating the effect of this treatment on individual outcomes. Assume that the underlying model is yigt = cgt + εigt . IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 17 The group error term, cgt , and the individual error term, εigt , are both independent normals with variance σc2 and σε2 . Take σc2 = 0.1 and σε2 = 1. Note that the result is not sensitive to this choice. I experiment with different numbers of of groups (G) and different number of time periods (T ). Data are generated with a constant group-time cell size, ngt = n. In all experiments 50,000 simulations are performed. I estimate models of the form yigt = αg + αt + bDgt + cgt + εigt . This represents a general DID setting, with ﬁxed group effects, αg , ﬁxed time effect, αt , and a treatment indicator variable, Dgt , taking the value one if the treatment is imposed in group g at time point t. b is then the treatment effect. The treatment status is randomly assigned. In the basic case we take two time periods (T = 2) and two groups (G = 2). The treatment status is then assigned to one of the groups (G1 = 1), and they experience the treatment in the second period. Besides the basic case, other combinations of T ,G and D are considered 13 . To be precise, the basic model with T = 2, G = 2 and G1 = 1 includes two group dummies, one time dummy for the second period, and one treatment dummy taking the value one in the second period for group two. The models for other combinations of T ,G and D follow in the same way. 5.1 Small sample properties As shown in Section 3, the sensitivity analysis method can be used to derive a cut-off value, γc . This value can be seen as a test-statistic. If one is conﬁdent that this value is unreasonably large one should reject the null-hypothesis of zero effect. In other words the critical value is decided by the researchers knowledge about reasonable values of γ. If the researcher knows the true relation between σc2 and σε2 , referred to as γt = σc2 /σε2 , then theoretically if N is large a test for b = 0 using γc as a test-statistic and using γt as the critical value should have the correct size. This should hold for any combination of T ≥ 2,G ≥ 2 and G > G1 . This subsection conﬁrms this property. I also examine the small sample properties of this approach. To this end the approach is somewhat mod 13 If T > 2 the treatment occurs after T /2 − 0.5 if T is a odd number and after T /2 if T is an even number. 18 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups Table 1: Monte Carlo results for the small sample properties of the sensitivity analysis method. Group Size (n) 10 20 50 100 1000 G = 2, T = 2 G1 = 1 G = 3, T = 2 G1 = 1 G = 3, T = 3 G1 = 1 G = 5, T = 5 G1 = 2 0.0503 0.0496 0.0505 0.0484 0.0505 0.0492 0.0489 0.0516 0.0519 0.0504 0.0495 0.0512 0.0494 0.0501 0.0495 0.0504 0.0500 0.0500 0.0505 0.0494 Notes: Monte Carlo results for the treatment parameter which enters the model with a true coeﬃcient of b = 0. The model and the data generating process is described in detail in the text. Each cell in the table reports the rejection rate for 5% level tests using the sensitivity analysis γc as test-statistic, and γt as critical value. Test based on a tnGT −G−T . T is the number of time periods, G the number of groups, and G1 the number of groups who receives the treatment. The number of simulations is 50,000. iﬁed. Asymptotically (in N) the sensitivity analysis method can be based on a normal distribution, regardless of the distribution of the individual error, ε. If N is small but ε is normally distributed the analysis should be based on a t-distribution with nGT − G − T degrees of freedom. This follows since the t-statistic reported in equation (12) has an exact t-distribution instead of a normal distribution. Table 1 present the results from this exercise. Each cell of Table 1 represents the rejection rate under the speciﬁc combination of n,T ,G,D, and γt . As apparent from the table, the sensitivity analysis method works as intended for all sample sizes. It conﬁrms that the derived properties of the sensitivity analysis method are correct. This is not surprising since the sensitivity analysis is based on OLS estimates with well established properties. It does not however give evidence for an inferential method in a strict statistical sense as the exact value of the used critical value γt is not known in practice. In practice reasonable values of γt have to be assessed, for instance using other data sources. 5.2 Robustness and comparison with other inference methods The researcher may have information through other data sources, or for other outcomes, which enables a closer prediction of γt . However information that enables an exact esti mate of γt is not likely to be available. The second experiment therefore test the robustness of the results with respect to assessing an incorrect γt . Distinguish between the true ratio between the two error variances, γt and the ratio that the researcher thinks is the correct one, γr . If γc > γr the sensitivity analysis suggests rejecting the null-hypothesis of zero effect. If γt > γr this leads to over-rejection of the null-hypothesis. Here the severity of IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 19 this problem is tested. As a comparison the sensitivity analysis method is contrasted with other methods com monly used to perform inference. The other methods include OLS estimates without any adjustment of the standard errors, labeled OLS regular. Furthermore, OLS estimates with the commonly used Eicker-White heteroscedasticiy robust standard errors for grouped data. I either ”cluster” at group level or ”cluster” at the group-time level, i.e. variance matrices which is robust to within group correlation, and robust to within group-time cell correlation, respectively. These inference methods are labeled cluster group and cluster group-time. They are by far the most common ways of correcting for the use of individual data and outcomes that vary only on at group level. The general cluster formula is N − 1 C � C � �� C � � �� C � � Vˆcluster = ∑ XcXc ∑ XcuˆcuˆcXc ∑ XcXc N − K C − 1 c=1 c=1 c=1 where c indicates the cluster and C the number of clusters. Further, uˆc is a vector contain ing the OLS residuals, and Xc is a matrix containing the observations of the independent variables for the individuals in cluster c. Also note that a degrees of freedom correction is used. The tests are based on a tC−1 , i.e. tG−1 for clustering at the group level, and tGT −1 for clustering at the group-time level. The two-step estimator suggested by Donald & Lang (2007) is also considered. In the present case with explanatory variables which vary only at group-level, and in the absence of correlation over time in cgt , the ﬁrst step is aggregation at the group-time level. This gives y¯gt = αg + αt + β Xgt + cgt + ε¯gt , where y¯gt and ε¯gt are the group-time averages of yigt and eigt . The second step amounts to estimating this model using OLS. In the present case when both the error terms are inde pendent normals the resulting t-statistic for the hypothesis test of b = 0 has a t-distribution with GT − K degrees of freedom. The number of variables, K, is here G + T . The upper panel of Table 2 presents the results for the sensitivity analysis method, and the lower panel presents the results for the other four methods. Size is for 5% level 20 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups tests for the treatment parameter which enters the model with a true coefﬁcient of b = 0. Power is 5% level test versus the alternative that b = 0.1. In this analysis we take n = 200. Before interpreting these results note that the power should be compared for tests with the same nominal size. Furthermore, the terminology size and power for the sensitivity analysis method is not entirely correct from a statistical point of view. The sensitivity analysis gives a ”test-statistic” as a cut-off value, γc , but the cut-off value is decided by the researchers assessment of a reasonable size off γ. In that sense it is a test, which makes it reasonable to report the size and power. Also note that the main point of this section is to explore how sensitive the results are to the assessment of a reasonable size off γ. First, consider the performance of the other methods commonly used to perform infer ence. The results for the OLS estimates using regular standard errors and the two cluster formulas conﬁrm what has been found in earlier studies, see e.g. Bertrand et al. (2004), Donald & Lang (2007), Cameron et al. (2007), and Hansen (2007a). The regular uncor rected OLS estimates have large size distortions. The rejection rate for 5% level tests is 0.256 with G = 2, T = 2, G = 1. The two OLS cluster estimators also suffer from large size distortions. As expected these methods behave poorly if the number of groups is small: after all they were designed for the case with a large number of groups. If the number of groups is only moderately small, say G = 5 and T = 5, these tests perform somewhat better.14 Next, consider the performance of the Donald & Lang (2007) two step estimator. If the model is just-identiﬁed as in the case with G = 2 and T = 2 the test of the null hypothesis should be done using a t-statistic with zero degrees of freedom. In other words it is not possible to use this test for just-identiﬁed models. Next consider how the two step estimator performs if the groups become somewhat larger, but are still very small. The results in Column 2, 3 and 4 show that the DL estimator has correct size if the model is not just-identiﬁed. This conﬁrm the results in Donald & Lang (2007). However, if the number of groups is very small (Column 2 and 3) the power of the DL estimator is low. Let us compare these results with the results for the sensitivity analysis method, which 14 Notice that this experiment is set up with no correlation between the groups or over time in cgt . If that were the case we could expect these cluster estimators to perform even worse. The size distortions for G = 5 and T = 5 would then be likely to also be very large. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 21 Table 2: Monte Carlo results for the sensitivity analysis method when the true relation between the variance of the group-time error and the individual error is unknown. G = 2, T = 2 G1 = 1 G = 3, T = 2 G1 = 1 G = 3, T = 3 G1 = 1 G = 5, T = 5 G1 = 2 Size Power Size Power Size Power Size Power Sensitivity analysis γt = 0.010 γr = 0.005 γr = 0.008 γr = 0.009 γr = 0.010 γr = 0.011 γr = 0.012 γr = 0.015 γr = 0.020 0.111 0.069 0.057 0.050 0.044 0.038 0.024 0.011 0.671 0.587 0.562 0.531 0.504 0.481 0.411 0.314 0.112 0.068 0.058 0.048 0.041 0.036 0.024 0.012 0.778 0.701 0.677 0.657 0.627 0.609 0.539 0.431 0.108 0.066 0.058 0.050 0.044 0.038 0.024 0.011 0.870 0.816 0.794 0.779 0.758 0.739 0.675 0.575 0.112 0.069 0.059 0.050 0.044 0.037 0.024 0.012 1.000 0.999 0.999 0.998 0.998 0.998 0.996 0.992 OLS regular Cluster group-time Cluster group DL two step 0.256 1.000 1.000 n.a. 0.817 0.978 0.978 n.a. 0.257 0.562 0.276 0.051 0.889 0.934 0.684 0.148 0.260 0.370 0.280 0.051 0.945 0.944 0.750 0.462 0.257 0.133 0.107 0.050 1.000 0.973 0.996 0.998 Notes: Monte Carlo results for simulated data. The model and the data generating process is described in detail in the text. γt is the true relation between the variance of the group-time error and the individual error, and γr the assessed relation between these two variance. Further T is the number of time periods, G the number of groups, G1 the number of groups who receives the treatment, and ngt the sample size for each group-time cell. Size is for 5% level tests for the treatment parameter which enters the model with a true coeﬃcient of b = 0. Power is 5% level test versus the alternative that b = 0.1. The number of simulations is 50,000. uses γc as the test-statistic and γr as the critical value.15 First, consider the results when γt = γr , i.e. the researcher is able to correctly assess the size of within group correlation. As before the test has the correct size. Since the size of the test for the sensitivity analysis method and the DL method are the same for the results in Column 2-4, the power esti mates are comparable. The results show that the power is higher in the sensitivity analysis method. If the number of groups is very small, as in Column 2 and 3, the difference is large. For example if G = 3, T = 2, and G = 1 the power is 0.657 for the sensitivity anal ysis compared to 0.148 for the DL two step estimator. If the number of groups becomes somewhat larger as in Column 4 the difference is smaller. In this case the Donald & Lang (2007) two step estimator is likely to be preferable to sensitivity analysis. Also note that even if G = 2, T = 2, G1 = 1 the power of the sensitivity analysis test is high. The previous comparison was based on the assumption that the researcher is able to 15 I will use the terminology ”size” and ”test” here even though the sensitivity analysis method is not a statis tical test, as the critical value is assessed and not calculated. 22 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups assess the correct value of γ. In practice this is unreasonable. It is therefore also interesting to see what happens if γt not equal to γr , i.e. when the researcher is unable to exactly infer γ. These results are also presented in Table 2. These results show that the sensitivity analysis method performs well if the difference between γr and γt is rather small. For example, the rejection rate for 5% level tests is 0.069 if γr = 0.008 and γt = 0.010. This is only a small over-rejection of the null-hypothesis. However if the difference between γr and γt becomes large, there are as expected substantial size distortions. To summarize, the Monte Carlo simulations have conﬁrmed that the derived proper ties of the sensitivity analysis method are correct for both large and small sample sizes. They further show that existent inference methods run into problem when the number of groups is very small. Finally, the results show that the sensitivity analysis method is applicable, even if the number of groups is very small, as long as the size of the within group correlation can be reasonably assessed. The two applications provided in the next section show that this can often can be done. 6 Applications16 6.1 Application 1: Disability beneﬁts Meyer et al. (1995)17 (MVD) study the effects of an increase in disability beneﬁts (work ers compensation) in the state of Kentucky. Workers compensation programs in the USA are run by the individual states. Here we describe some of the main features of the system in Kentucky. A detailed description is found in MVD. The key components are payments for medical care and cash beneﬁts for work related injuries. MVD focus on temporary beneﬁts, the most common cash beneﬁt. Workers are covered as soon as they start a job. The insurance is provided by private insurers and self-insurers. The insurance fees that 16 This section presents two different applications. In order to focus on the application of the sensitivity analysis approach we re-examine some basic results from the two studies. I should however point out that a more elaborated analysis is performed in both studies is. It includes estimating for different sample, different outcomes and including additional control variables. However the basic regressions re-examined here constitute an important part of both studies. 17 This data has also been reanalyzed by Athey & Imbens (2006). They consider non-parametric estimation, and inference under the assumption of no cluster effects. Meyer et al. (1995) also consider a similar reform in Michigan. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 23 employers pay are experience rated. If eligible the workers can collect beneﬁts after a seven day waiting period, but beneﬁts for these days can be collected retroactively if the duration of the claim exceeds two weeks. The claim duration is decided mainly by the employee and his or her doctor, and there is no maximum claim duration. The replacement rate in Kentucky before 1980 was 66 23 % and the beneﬁts could be collected up to a maximum of $131 per week. The reform as of July 15, 1980, analyzed by MVD increased the maximum level to $217 per week: a 66% increase or 52% over one year in real terms.18 The replacement rate was left unchanged. Thus workers with previous high earnings (over the new maximum level) experience a 66% increase in their beneﬁts, while the beneﬁts for workers with previous low earnings (below the old ceiling) were unchanged. This creates a natural treatment group (high earners) and a natural con trol group (low earners). MVD analyze the effect of the increase using a DID estimator, which contrasts the difference in injury duration between before and after the reform for the treatment group and the control group. The upper panel of Table 3 restates MVD’s results for the outcome mean log injury duration, taken from their Table 4.19 Column 1-4 present the pre-period and post-period averages for the treatment and control group, Column 5 and 6 the difference between the pre- and post-period for the two groups, and Column 7 present the DID estimate. The DID estimate of the treatment effect is statistically signiﬁcant and suggests that the increased beneﬁts increased the injury duration by about 19%. MVD ignores the cluster sample issue and use regular OLS standard errors. Thus their standard errors are biased downwards if there are any cluster effects. It is also not possible to perform Donald & Lang (2007) inference, since the model is just-identiﬁed.20 It is also clear that MVD study an interesting question, and we ultimately want to learn something from the reform in Kentucky. The study by MVD is therefore a good example where sensitivity analysis should be applied. 18 For calculations see Meyer et al. (1995) p 325. terminology ”mean” is not totally accurate. The outcome used by MVD is censored after 42 months. However, at this duration only about 0.5% of the cases are still open. MVD therefore sets all ongoing spells to 42 months. Meyer et al. (1995) also consider other outcome variables and note that their results are quite sensitive to the choice of speciﬁcation. Here, the focus is on their preferred outcome. 20 The model includes four variables: a constant, a group dummy, a time dummy and a group time interaction. 19 The 24 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups Table 3: Sensitivity analysis estimates for application 1 on disability beneﬁts Log duration Sample Size Treated (High earnings) Pre Post period Period [1] [2] Non-Treated (Low earnings) Pre Post period Period 3] [4] 1.38 (0.04) 1,233 1.13 (0.03) 1,705 Sensitivity Analysis: γc - 5 % [10%] √ γc ∗ σε : - 5 % [10%] 1.58 (0.04) 1,161 1.13 (0.03) 1,527 Diﬀerences DID [2-1] [4-3] [5-6] [5] [6] [7] 0.20 (0.05) 0.01 (0.04) 0.19 (0.07) 0.0026 [0.0041] 0.0629 [0.0787] - 0.00067 [0.00127] 0.0335 [0.0461] - Notes: The results in the upper panel are taken from Meyer et al. (1995), their standard errors in parentheses. The outcome is mean log duration, censored after 42 months. The sensitivity analysis results in the lower panel is own calculations. γc is calculated by numerically solving for γ in equation (16), for the speciﬁed signiﬁcance level. Let us start with the basic sensitivity analysis, applicable under the most restrictive assumptions, namely that the cluster-effects (group-time speciﬁc shocks) are uncorrelated between the groups as well as uncorrelated over time. The sensitivity analysis presented in Section 3 is then applicable. The γc values for 5% level (10-% in brackets) under these assumptions are reported in the lower panel of Table 3. We report cut-off values for both the difference estimates as well as the DID estimate.21 The 5% level cut-off value for the DID estimate is 0.00067. The meaning of this estimate is that the variance of the group time shocks is allowed to be 0.00067 times the variance of the unobserved individual heterogeneity before the treatment effect is rendered insigniﬁcant. At ﬁrst glance it may seem difﬁcult to assess whether this is a unreasonably large value. Table 3 therefore also reports these values recalculated into cut-off standard deviations for the group-time shocks √ ( γc σε ). These cut-off values show that the standard deviation of the group-shocks is allowed to be 0.034 on 5% level (0.046 10% level). Column 1 and Column 3 show that the mean of the outcome log injury duration are 1.38 and 1.13 for the treatment group and the control group before the reform. Compared to these means the allowed standard deviation of the shocks is quite large. Furthermore, Column 6 show that the change in injury duration in the control group between the two time periods is 0.01. Even if if does 21 Notice that no cut-off values are reported for the control group since the difference for this group is already insigniﬁcant using the regular standard errors. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 25 not offer conclusive evidence, it suggests that the variance of the group-time shocks is small. Taken together it is therefore fair to say that there is a statistically signiﬁcant effect on the injury duration. Next consider an extended sensitivity analysis, which allows for correlation over time in the group-time shocks. In order to take this into account, replace the assumption of no autocorrelation in the cluster effects with an assumption of ﬁrst order autocorrelation in these shocks. This gives two sensitivity parameters, γ and κ, measuring the size of the cluster effects and the correlation over time in these cluster effects. Since MVD work with repeated cross-section data we can directly apply the results in subsection 4.1. The results from this exercise are presented in Figure 1, displaying cut-off values at 10% level for standard deviation of the group-speciﬁc time for a range of κ values. In this case with two time periods, a positive autocorrelation in the group-time shocks increases the cut-off values for γ. This extended sensitivity analysis therefore ultimately strengthening the conclusion that there is a statistical signiﬁcant effect on the injury duration from an increase in disability beneﬁts. Figure 1: Two parameter sensitivity analysis for the DID estimates in Meyer et al. (1995). Autocorre lation in group-time shocks and allowed standard deviation of the group-time shocks. 6.2 Application 2: Earned income tax credit Eissa & Liebman (1996)(EL) study the impact of an expansion of the Earned income tax credit (EITC) in the USA on the labor force participation of single women with children. EITC was introduced in 1975. Currently a taxpayer needs to meet three requirements in 26 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups order to be eligible for the tax credit. The taxpayer need to have positive earned income, the gross income must be below a speciﬁed amount, and ﬁnally the taxpayer needs to have a qualifying child.22 The amount of the credit is decided by the taxpayers earned income. The credit is phased in at a certain rate for low incomes, then stays constant within a certain income bracket, and is phased out at a certain rate for higher earnings. High earners are therefore not entitled to any EITC tax credit. EL study the effects of the 1987 expansion of EITC in USA on labor supply. The re form changed EITC in several ways. The main changes were increases in the subsidy rate for the phase-in of the credit, an increase in the maximum income to which the subsidy rate is applied, and a reduction in the phaseout rate. This resulted in an increase in the maximum credit from $550 to $851, and made taxpayers with income between $11,000 and $15,432 eligible for the tax credit. All these changes made EITC more generous and the treatment consist of the whole change in the budget constraint. Obviously the reform only changes the incentives for those eligible for the tax credit. One key requirement is the presence of a qualifying child in the family. A natural treatment group is then sin gle women with children, and a natural control group is single women without children. However, some single women with children are high income earners and thus are most likely to be unaffected by the EITC reform. EL therefore further divides the sample by education level. Here we report the results for all single women and single women with less than high-school education, from now on referred to as low educated. EL use CPS data to estimate the treatment effect. Their outcome variable is an indica tor variable taking the value one if the annual hours worked is positive. Similarly to MVD they use a DID approach, which contrast the differences between the post- and pre-reform period labor supply for the treatment and the control group. The main results from their analysis are presented in the upper panel of Table 4, taken from Table 2 in EL. The results from the DID analysis, presented in Column 7, suggest a positive and statistically signiﬁ cant effect of the EITC expansion in both speciﬁcations. If all single women are used, EL estimates that the expansion increased the labor force participation with 2.4 percentage points (4.1 percentage points for low educated single women). 22 A qualifying child is deﬁned as a child, grandchild, stepchild, or foster child of the taxpayer. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 27 Table 4: Sensitivity analysis estimates for application 2 on earned income tax credit Sample All Low education Sample Size All Low education Treated (with children) Pre Post period Period [1] [2] Non-Treated (without children) Pre Post period Period 3] [4] 0.729 (0.004) 0.479 (0.010) 0.952 (0.001) 0.784 (0.010) 20,810 5396 Sensitivity Analysis: γc - 5 % [10%] All Low education √ γc ∗ σε : - 5 % [10%] All Low education 0.753 (0.004) 0.497 (0.010) 0.952 (0.001) 0.761 (0.009) Diﬀerences DID [2-1] [4-3] [5-6] [5] [6] [7] 0.024 (0.006) 0.018 (0.014) 0.000 (0.002) -0.023 (0.013) 0.024 (0.006) 0.041 (0.019) 0.00030 [0.00048] - - 0.00022 [0.00034] 0.00005 [0.00031] 0.0075 [0.0094] - - 46,287 3958 - - 0.0053 [0.0066] 0.0043 [0.0080] Notes: The results in the upper panel are taken from Eissa & Liebman (1996), their standard errors in parentheses. The outcome is an indicator variable taking the value one is hours worked is positive, and zero otherwise. Two diﬀerent samples, all single women and single women with less than high school. The sensitivity analysis results in the lower panel is own calculations. γc is calculated by numerically solving for γ in equation (16), for the speciﬁed signiﬁcance level. The calculations are made under the assumption that the sample size is the same before and after the reform in the two groups. The inference issues are very similar to those of the MVD study. In the presence of any group-time effects the standard errors presented by EL are biased downwards. We have two DID models, which both are just-identiﬁed, making sensitivity analysis an attractive alternative. I ﬁrst consider sensitivity analysis under assumption of no autocorrelation in the group-time shocks, and then we allow for ﬁrst order autocorrelation in these shocks. The results from the basic sensitivity analysis is presented in the lower panel of Table 4. The 5 percent, γc , cut-off value for the two DID estimates is 0.00022 for the full sample and 0.00005 for the sample of low educated mothers. It implies that the variance of the group-time shocks is allowed to be 0.0002 and 0.00005 times the variance of the unobserved individual heterogeneity. It further means that the standard deviation of the group-time shocks is allowed to be about 0.005 for the full sample and about 0.004 for the smaller sample of low educated mothers. In other words even very small shocks 28 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups render the treatment effect insigniﬁcant. It can be compared with the mean labor force participation before the reform, which was 0.73 for all single women with children and 0.48 for low educated single mothers. Single women with children are after all a quite different group compared to single women without children. We can therefore expect quite large group-time speciﬁc shocks. Furthermore, there is a large drop of 0.023 in the labor force participation for the control group of low educated single women without children. It therefore seems unreasonable to believe that the variance of shocks is smaller than the variance implied by the cut-off values. Figure 2: Two parameter sensitivity analysis for the DID estimates in Eissa & Liebman (1996). Left panel: the full sample of single women and right panel: the sample of low educated single women. Autocorrelation in group-time shocks and allowed standard deviation of the group-time shocks. Next consider allowing for ﬁrst order autocorrelation in the group-time effects. As in the previous application we use the results in subsection 4.1 for repeated cross-section data. The cut-off standard deviation of the group shocks at 10% level is displayed for a range of κ values in Figure 2. The left graph display the cut-off values for the full sample and the right graph displays the cut-off values for the smaller sample of low educated mothers. Introducing autocorrelation in the two group two time period case increases the allowed variance of the group speciﬁc shocks. However, the variance is still only allowed to be very small before the estimates are rendered insigniﬁcant. We therefore conclude based on the estimates presented, that there is no conclusive evidence of any important labor supply effects from the EITC expansion in 1987. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 29 7 Conclusions Many policy analyses rely on variation at the group level to estimate the effect of a policy at the individual level. A key example used throughout this paper is the difference-in differences estimator. The grouped structure of the data introduces correlation between the individual outcomes. This clustering problem has been addressed in a number of different studies. In this paper I have introduced a new method of perform inference when faced with data from only a small number of groups. The proposed sensitivity analysis approach is even able to handle just-identiﬁed models, including the often used two group two time period difference-in-differences setting. Consider for example having data for men and women, for two cities or for a couple of villages. The key feature of the proposed sensitivity analysis approach is that all focus is placed on the size of the cluster effects, or simply the size of the within group correlation. Pre viously in the applied literature a lot of discussion concerned no within group correlation against non-zero correlation, since these two alternatives imply completely different ways to perform inference. This is a less fruitful discussion. In the end it is the size of the clus ter effects which matters. In some cases it is simply not likely to believe that an estimated treatment effect is solely driven by random shocks, since it would require these shocks to have a very large variance. The sensitivity analysis formalizes this discussion by assessing how sensitive the standard errors are to within-group correlation. In order to demonstrate that our method really can distinguish a causal effect from random shocks the paper offered two different applications. In both applications key re gressions are based on just-identiﬁed models. The sensitivity analysis results indicate that the conclusion from the ﬁrst study that the treatment effect is signiﬁcant is not sensitive to departure from the independence (no-cluster) assumption, whereas the results of the second study are sensitive to the same departure and its conclusion cannot therefore be trusted. More precisely in one of the applications it is not likely that the group effects are so large in comparison with the individual variation that it would render the estimated treatment effect insigniﬁcant. In the second application even small within group correla tion renders the treatment effect insigniﬁcant. 30 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups Besides offering a new method of performing inference, this paper contributes by introducing a new type of sensitivity analysis. Previously in the sensitivity analysis liter ature, the sensitivity of the point estimate has been investigated. This paper shows that sensitivity analysis with respect to bias in the standard errors may be equally important. This opens a new area for future sensitivity analysis research. IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 31 References Abadie, A. 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Start with equation (8) V = N −K (X � X)−1 X �CXE(Vˆ ). N − tr[(X � X)−1 X �CX] First, consider (X � X)−1 X �CX. Under assumption of xigt = xgt we have ⎡ x1 ⎤ ⎡ ⎢ ⎥ ⎢ ⎥ ⎢ x2 ⎥ ⎥ X = ⎢ ⎥ ⎢ . . ⎢ . ⎥ ⎣ ⎦ xG � lg1 xg1 ⎤ ⎢ ⎥ ⎢ ⎥ � ⎢ lg2 xg2 ⎥ ⎢ xg = ⎢ . ⎥ ⎥ ⎢ . . ⎥ ⎣ ⎦ � lgT xgT , where lgt is a column vector of ngt ones, G is the number of groups and T is the number of time periods. If E(cgt cg�t ) = 0 for all t and all g �= g� , and E(cgt cgt � ) = 0 for all g and all t �= t � , we further have ⎡ ⎤ ⎡ ⎤ C . . . 0 C . . . 0 ⎢ 1 ⎢ g1 ⎥ ⎥ . . . ⎥ ⎢ . . . . ⎢ ⎥ .. C = ⎢ . . . . ⎥ Cg = ⎢ . . . . . ⎥ ⎣ ⎣ ⎦ ⎦ 0 . . . CG 0 . . . CgT with ⎡ 1 ⎢ ⎢ ⎢p Cgt = ⎢ ⎢ .. ⎢. ⎣ p ⎤ . . . p ⎥ .. ⎥ 1 .⎥ � ⎥ . . . ⎥ = [(1 − p)Igt + plgt lgt ] p⎥ ⎦ ... p 1 p Here Igt is a unit matrix of order ngt , and p ≡ σc2 2 σc +σε2 . It follows that � X � X = ∑∑ ngt xgt xgt g 36 (A.1) t IFAU – Cluster sample inference using sensitivity analysis: the case with few groups and � � Cgt lgt xgt . X �CX = ∑ ∑ xgt lgt g (A.2) t and ⎡ ⎤ 1 + (ngt − 1)p ⎢ ⎥ ⎢ ⎥ ⎢1 + (ngt − 1)p⎥ � � � � � ⎢ ⎥ x gt = xgt ngt [1 + (ngt − 1)p]xgt xgt lgt Cgt lgt xgt = xgt lgt ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ 1 + (ngt − 1)p (A.3) Combining equation (A.1), (A.2) and (A.3) gives � −1 � X � XX �CX = (∑ ∑ ngt xgt xgt ) ∑ ∑ ngt τgt xgt xgt g t g (A.4) t with τgt = 1 + (ngt − 1)p. Imposing ngt = n we have equation (A.4) as X � XX �CX = τIK (A.5) with τ = 1 + (n − 1)p. Next consider N−K : N−tr[(X � X)−1 X �CX] using the result in equation (A.5) gives tr[(X � X)−1 X �CX] = Kτ. (A.6) Substituting equation (A.5) and equation (A.6) into equation (8) and imposing ngt = n (then N = nGT ) gives nGT − K V = E(Vˆ )τ , nGT − Kτ i.e. equation (9). IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 37 Derivation of equation 18. Again start with equation (8) V= N −K (X � X)−1 X �CXE(Vˆ ). � −1 � N − tr[(X X) X CX] First, consider (X � X)−1 X �CX. Remember that C is deﬁned as E(ee� ) = σ 2C, where e is a vector collecting all eigt = cgt + εigt , and σ 2 ≡ 1/Ntr(ee� ). In order to ex press C in terms of κ and γ use the well know properties of an AR(1) process (under the assumption of |κ| < 1), and the deﬁnition of σd2 ≡ γσε2 from section 4.1. This gives σc2 σd2 γσε2 = E(cgt cgt ) = = 1 − κ2 1 − κ2 (A.7) and if t �= t � � Cov(cgt cgt � ) = E(cgt cgt � ) = κ |t−t | 2 σd2 |t−t � | γσε = κ . 1 − κ2 1 − κ2 (A.8) Thus if i = j E(eigt e jgt ) = σ 2 = σc2 + σε2 = 2 γσε2 2 21+γ −κ = σ + σ . ε ε 1 − κ2 1 − κ2 (A.9) Further, using (A.7) and (A.9), if i = � j γ γσε2 = σ2 2 1−κ 1 + γ − κ2 (A.10) σd2 � γ = σ 2 κ |t−t | . 2 1−κ 1 + γ − κ2 (A.11) E(eigt e jgt ) = σc2 = and using (A.8) and (A.9), if t �= t � � E(eigt e jgt � ) = κ |t−t | 38 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups and under assumption E(cgt cgt � ) = 0, if g �= g� E(eigt e jg�t � ) = 0 (A.12) Then under (A.9), (A.10), (A.11) and (A.12) ⎡ ⎤ ⎡ ⎤ C11 . . . CT 1 C1 . . . 0 ⎥ ⎢ ⎥ ⎢ . ⎥ . ⎥ ⎢ . ⎢. C = ⎢ . . . . . . . ⎥ Cg = ⎢ . . . . . . . ⎥ ⎦ ⎣ ⎦ ⎣ 0 . . . CG C1T . . . CT T (A.13) γ γ )Igt + lgt l � ] 2 1 + γ − κ 2 gt 1+γ −κ (A.14) γ lgt l � . 1 + γ − κ 2 gs (A.15) with if t = t � Ctt � = [(1 − and if t �= t � � Ctt � = κ |t−t | Deﬁne pc = γ . 1+γ−κ 2 Then, using equation (A.14), if t = t � ⎡ ⎤ 1 + (ngt − 1)pc ⎢ ⎥ ⎢ ⎥ ⎢1 + (ngt − 1)pc ⎥ � � � ⎢ � � ⎥ x gt = xgt ngt [1 + (ngt − 1)pc ]xgt , xgt lgt Ctt lgt xgt = xgt lgt ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ 1 + (ngt − 1)pc (A.16) and, using equation (A.15), if t = � t� � � xgt lgt Ctt � lgt � xgt � ⎡ ⎤ � n p ⎢ gt c ⎥ ⎥ ⎢ ⎢n � p ⎥ � � ⎢ gt c ⎥ � |t−t � | � xgt lgt ⎢ . ⎥ x gt � = κ |t−t | xgt ngt pc ngt � xgt =κ � . ⎢ . ⎥ ⎦ ⎣ ngt � pc (A.17) Using equation (A.1), (A.16) and (A.17) gives � −1 ) X � XX �CX = (∑∑ ngt xgt xgt g (A.18) t IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 39 � � � ( ∑ ∑ ∑ (κ |t−t | ngt ngt � xgt xgt � ) + ngt [1 + (ngt − 1)pc ]xgt xgt ) g t t � �=t Imposing ngt = n, substituting for pc = γ 1+γ−κ 2 and simplifying we have equation (A.18) as X � XX �CX = (1 + (n − 1) γ γ )IK + n H 2 1 + γ − κ2 1+γ −κ (A.19) with � � −1 � ) ∑ ∑ ∑ (κ |t−t | xgt xgt H = ( ∑ ∑ xgt xgt � ). g Next consider t g N−K : N−tr[(X � X)−1 X �CX] t t � �=t using the results in (A.19) gives tr[(X � X)−1 X �CX] = K(1 + (n − 1) K γ γ ) + n ∑ Haa. 1 + γ − κ2 1 + γ − κ 2 a=1 (A.20) where Haa is the element in the ath column and ath row of H. Substituting equation (A.19) and equation (A.20) into equation (8) and noting that under ngt = n we have N = nGT gives Vaa = E(Vˆaa ) nGT − K γ γ K nGT − K(1 + (n − 1) 1+γ−κ 2 ) + n 1+γ−κ 2 ∑a=1 Haa (1 + (n − 1) γ γ ) + n Haa 1 + γ − κ2 1 + γ − κ2 Both the ﬁrst and the second part of this expression, the two sources of bias in the stan dard errors are greater than one. However, it will be highly dominated by (1 + (n − γ γ 1) 1+γ−κ 2 )IK + n 1+γ−κ 2 Haa . Thus we have Vaa ≈ E(Vˆaa )(1 + (n − 1) γ γ )+n Haa . 2 1+γ −κ 1 + γ − κ2 i.e. equation (18). Derivation of equation 22. Again start with equation (8) V= 40 N −K (X � X)−1 X �CXE(Vˆ ). � −1 � N − tr[(X X) X CX] IFAU – Cluster sample inference using sensitivity analysis: the case with few groups Using the deﬁnition σc2 ≡ γσε2 from section 4.1, if i = j we have E(eigs e jgs ) = σ 2 = σc2 + σε2 = σε2 (1 + γ). (A.21) Using this and under assumption E(cgt cg�t ) = 0 for all t, and the multiway clustering assumptions if s = � s� then E(cgs cg� s� ) = 0 and if s = s� then E(cgs cg� s� ) = ξ σc2 , it gives if i= � j E(eigs e jgs ) = σc2 = γσε2 = σ 2 γ 1+γ (A.22) and if i �= j and g = g� holds E(eigs e jg� s ) = ξ σc2 = ξ γσε2 = σ 2 ξ γ 1+γ (A.23) and if s �= s� E(eigs e jg� s ) = 0. (A.24) Thus under (A.21), (A.22), (A.23) and (A.24) ⎡ C . . . ⎢ 1 ⎢. C = ⎢ . . . . . ⎣ 0 ... ⎤ ⎡ ⎤ 0 C11 . . . CGs 1 ⎥ ⎢ ⎥ . ⎥ ⎢ . . .. . . . ⎥ . = C . ⎥ s ⎢ . . ⎥ ⎦ ⎣ ⎦ CS C1Gs . . . CGs Gs (A.25) with if g = g� Cgg� = [(1 − γ γ )Ig + lg ls ] 1+γ 1+γ (A.26) and if g �= g� Cgg� = ξ γ lg lg� . 1+γ (A.27) Here Gs is the number of groups belonging to group-cluster s. Retain the deﬁnition p = IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 41 γ 1+γ , and deﬁne lgs as a column vector of ngs ones. Then, using equation (A.26), if g = g� ⎡ ⎤ 1 + (ngs − 1)p ⎢ ⎥ ⎢ ⎥ ⎢1 + (ngs − 1)p⎥ � � � � ⎢ � ⎥ x gs = xgs ngs [1 + (1 − ngs )p]xgs , xgs lgsCgg lgs xgs = xgs lgs ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ 1 + (ngs − 1)p (A.28) and, using equation (A.27), if g �= g� ⎡ ⎤ ng� s p ⎢ ⎥ ⎢ ⎥ ⎢n � p⎥ � � � ⎢ gs ⎥ � xgs lgsCgg� lg� s xg� s = ξ xgs lgs ⎢ . ⎥ x g� s = ξ xgt ngs pc ng� s xg� � s ⎢ .. ⎥ ⎣ ⎦ ng� s p (A.29) Using equation (A.1), (A.28) and (A.29) gives � −1 ) X � XX �CX = (∑ ∑ ngs xgs xgs s ( ∑ ∑ s (A.30) g � (ξ ngs ng� s xgs xg� � s ) + ngs [1 + (1 − ngs )p]xgs xgs ) ∑ � g g �=g Imposing ngs = n, substituting for p = γ 1+γ and simplifying we have equation (A.30) as X � XX �CX = (1 + (n − 1) γ γ )IK + n ξ M, 1+γ 1+γ (A.31) with � −1 ) ∑∑ M = ( ∑ ∑ xgs xgs s Next consider g N−K , N−tr[(X � X)−1 X �CX] s (xgs xg� � s ). ∑ � g g �=g using the results in (A.31) gives tr[(X � X)−1 X �CX] = K(1 + (n − 1) K γ γ ) + n ξ ∑ Maa . 1 + γ 1 + γ a=1 (A.32) where Maa is the element in the ath column and ath row of M. 42 IFAU – Cluster sample inference using sensitivity analysis: the case with few groups Substituting equation (A.31) and equation (A.32) into equation 8 and noting that under ngt = n we have N = nGT gives Vaa = E(Vˆaa ) nGT − K γ γ nGT − K(1 + (n − 1) 1+γ ) + n 1+γ ξ ∑K a=1 Maa (1 + (n − 1) γ γ +n ξ Maa ). 1+γ 1+γ Both the ﬁrst and the second part of this expression, the two sources of bias in the standard errors, are greater than one. However, it will be highly dominated by (1 + (n − 1) 1+γ γ + n 1+γ γ ξ Maa ). Thus we have Vaa ≈ Vˆaa (1 + (n − 1) γ γ +n ξ Maa ) 1+γ 1+γ i.e equation (22). IFAU – Cluster sample inference using sensitivity analysis: the case with few groups 43 Publication series published by the Institute for Labour Market Policy Evaluation (IFAU) – latest issues Rapporter/Reports 2009:1 Hartman Laura, Per Johansson, Staffan Khan and Erica Lindahl, ”Uppföljning och utvärdering av Sjukvårdsmiljarden” 2009:2 Chirico Gabriella and Martin Nilsson ”Samverkan för att minska sjukskrivningar – en studie av åtgärder inom Sjukvårdsmiljarden” 2009:3 Rantakeisu Ulla ”Klass, kön och platsanvisning. 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